The present invention relates to lithographic methods, and particularly to a method for matching a plurality of exposure tools using a programmable illuminator.
Lithographic exposure tools can typically provide disk, dipole, and quadrupole sources, as well as annuli, and along the above lines one can sample the available parametric options for all of these shape choices in such a way as to add to one's source pixel collection a close approximation to all disk, dipole, and quadrupole sources that the tool can provide. To design a multiscan lithographic source formed by these standard sources, one needs to determine the appropriate intensity value that should be assigned to the illuminating beams corresponding to every pixel in the collection.
According to prior art source optimization, linear programming (“LP”) algorithm can be employed to calculate the optimal intensity values for such a collection of source pixels. In this case, the optimal solution typically contains a fairly large number of non-zero members. However, it is generally desirable that a multiscan source be composed of only a small number of component scans, for example, three or less, in order to prevent the multiple-scan exposure process from becoming inordinately slow. If the photoresist must remain sensitized (pre-bake-and-develop) during more than three scans, the properties of the photoresist may begin to deteriorate. In addition, the throughput of the exposure tool may be severely constricted.
The prior art source optimization optimizes the pixel configuration of the source by maximizing an objective function ψ that is a metric of the quality of an exposure tool's lithographic printing, such as process window. This method is a global optimization in which the starting point of the solution is not predefined. Using notations in A. E. Rosenbluth and N. Seong, “Global Optimization of the Illumination Distribution to Maximize Integrated Process Window,” SPIE v.6154 Optical Microlithography XIX (2006) and U.S. Pat. No. 7,057,709 to A. E. Rosenbluth, “Printing a Mask with Maximum Possible Process Window Through Adjustment of the Source Distribution,” which are incorporated herein by reference, an embodiment using LP has the following form:
            Maximize              w        .        r        .        t        .                                  ⁢                      ⁢          {              ψ        ⁡                  (                                    z              o                        ,                          K              Min                        ,                                          K                Max                            ;                                              )                    ]        ,          ⁢      (          note      ⁢                          ⁢                                                 ``                    ⁢          w                .        r        .        t        ⁢                  .          ″                ⁢                                  ⁢                              means            ⁢                                                          ``                    ⁢      with      ⁢                          ⁢      respect      ⁢                          ⁢              to        ⁢                  .          ″                      )  
where objective ψ is defined by:
            ψ      ⁡              (                              z            0                    ,                      K            Min                    ,                                    K              Max                        ;                                      )              ≡                  ∑                  k          =                      K            Min                                    K          Max                    ⁢              (                              w            k            ′                    -                      w            k                          )              ,subject to:
                                              ⁢                  (                      eq            .                                                  ⁢            1                    )                                                                                              ⁢                  0          ≤                                                  j                        ⁢                          S              Min                                ≤                                    S                              Max                ,                j                                      ⁢                                        ·                              p                ->                                      ⁢                                                  ⁢                          (                                                ∀                  j                                |                                  1                  ≤                  j                  ≤                                      J                    Max                                                              )                                                          a        )                                                          ⁢                                          ·                                                            I                  ->                                                  i                  ⁡                                      [                                          r                      ,                      n                                        ]                                                  ′                            ⁡                              (                                  0                  ,                                      z                    0                                                  )                                              =                      1            ⁢                                                  ⁢                          (                                                ∀                  n                                ,                                  r                  |                                      1                    ≤                    r                    ≤                                                                  r                                                  Max                          ,                          OPC                                                                    ⁡                                              [                        n                        ]                                                                                            ,                                  1                  ≤                  n                  ≤                                      n                    Max                                                              )                                                          b        )                                                    ·                                                    I                ->                                            i                ⁡                                  [                                      u                    ,                    n                                    ]                                            ′                        ⁡                          (                              0                ,                                  z                  0                                            )                                      ≥                              R            Bright                          (              n              )                                ⁢                                          ⁢                      (                                          ∀                n                            ,                              u                |                                  1                  ≤                  u                  ≤                                                            u                      Max                                        ⁡                                          [                      n                      ]                                                                                  ,                              1                ≤                n                ≤                                  n                  Max                                                      )                                              c        )                                                    ·                                                    I                ->                                            i                ⁡                                  [                                      v                    ,                    n                                    ]                                            ′                        ⁡                          (                              0                ,                                  z                  0                                            )                                      ≤                              R            Dark                          (              n              )                                ⁢                                          ⁢                      (                                          ∀                n                            ,                              v                |                                  1                  ≤                  v                  ≤                                                            v                      Max                                        ⁡                                          [                      n                      ]                                                                                  ,                              1                ≤                n                ≤                                  n                  Max                                                      )                                              d        )                                          w          k                ≥                                          ·                                                            I                  ->                                                  i                  ⁡                                      [                                          r                      ,                      n                                        ]                                                  ′                            ⁡                              (                                                      CD                    +                                    ,                                                            z                      0                                        +                                          k                      ⁢                                                                                          ⁢                      Δ                      ⁢                                                                                          ⁢                      z                                                                      )                                              ⁢                                          ⁢                      (                                          ∀                r                            ,              n              ,                              k                |                                  1                  ≤                  r                  ≤                                                            r                      Max                                        ⁡                                          [                      n                      ]                                                                                  ,                                                K                  Min                                ≤                k                ≤                                  K                  Max                                            ,                              1                ≤                n                ≤                                  n                  Max                                                      )                                              e        )                                          w          0          ′                ≥                                          ·                                                            I                  ->                                                  i                  ⁡                                      [                                          r                      ,                      n                                        ]                                                  ′                            ⁡                              (                                                      CD                    +                                    ,                                                            z                      0                                        +                                          k                      ⁢                                                                                          ⁢                      Δ                      ⁢                                                                                          ⁢                      z                                                                      )                                              ⁢                                          ⁢                      (                                          ∀                r                            ,              n              ,                              k                |                                  1                  ≤                  r                  ≤                                                            r                      Max                                        ⁡                                          [                      n                      ]                                                                                  ,                                                K                  Min                                ≤                k                ≤                                  K                  Max                                            ,                              1                ≤                n                ≤                                  n                  Max                                                      )                                              f        )                                          w          k          ′                ≤                                          ·                                                            I                  ->                                                  i                  ⁡                                      [                                          r                      ,                      n                                        ]                                                  ′                            ⁡                              (                                                      CD                    -                                    ,                                                            z                      0                                        +                                          k                      ⁢                                                                                          ⁢                      Δ                      ⁢                                                                                          ⁢                      z                                                                      )                                              ⁢                                          ⁢                      (                                          ∀                r                            ,              n              ,                              k                |                                  1                  ≤                  r                  ≤                                                            r                      Max                                        ⁡                                          [                      n                      ]                                                                                  ,                                                K                  Min                                ≤                k                ≤                                  K                  Max                                            ,                              1                ≤                n                ≤                                  n                  Max                                                      )                                              g        )                                          w          0                ≤                                          ·                                                            I                  ->                                                  i                  ⁡                                      [                                          r                      ,                      n                                        ]                                                  ′                            ⁡                              (                                                      CD                    -                                    ,                                                            z                      0                                        +                                          k                      ⁢                                                                                          ⁢                      Δ                      ⁢                                                                                          ⁢                      z                                                                      )                                              ⁢                                          ⁢                      (                                          ∀                r                            ,              n              ,                              k                |                                  1                  ≤                  r                  ≤                                                            r                      Max                                        ⁡                                          [                      n                      ]                                                                                  ,                                                K                  Min                                ≤                k                ≤                                  K                  Max                                            ,                              1                ≤                n                ≤                                  n                  Max                                                      )                                              h        )                                                          ⁢                              w            k                    ≥                                    w                              k                -                1                                      ⁢                                                  ⁢                          (                                                ∀                  k                                |                                  1                  ≤                  k                  ≤                                      K                    Max                                                              )                                                          i        )                                                          ⁢                              w            k                    ≤                                    w                              k                -                1                                      ⁢                                                  ⁢                          (                                                ∀                  k                                |                                                                            K                      Min                                        +                    1                                    ≤                  k                  ≤                                      -                    1                                                              )                                                                    i          ′                )                                                          ⁢                              w            k            ′                    ≤                                    w                              k                -                1                            ′                        ⁢                                                  ⁢                          (                                                ∀                  k                                |                                  1                  ≤                  k                  ≤                                      K                    Max                                                              )                                                          j        )                                                          ⁢                              w            k            ′                    ≥                                    w                              k                -                1                            ′                        ⁢                                                  ⁢                          (                                                ∀                  k                                |                                                                            K                      Min                                        +                    1                                    ≤                  k                  ≤                                      -                    1                                                              )                                                                    j          ′                )                                                          ⁢                              w            k                    ≤                                    w              k              ′                        ⁢                                                  ⁢                          (                                                ∀                  k                                |                                                      K                    Min                                    ≤                  k                  ≤                                      K                    Max                                                              )                                                          k        )            
The script s, w, and w′ variables are always allowed to vary in the above prior art LP formulation. This LP formulation can optionally be embedded in an outer loop or optimization in which z0, Kmin, and Kmax are allowed to vary, but these latter parameters are treated as constant when the above LP is solved.
The constraints in eq. 1 are only representative. Many different kinds of constraints can be used to define the set of possible (i.e., allowed) intensities at image sample points and in the source beams. The symbols used in the eq. 1 can readily be employed to define many variants, as will be clear to those skilled in the art.
Variables z0, KMax, and KMin are the centerpoint, upper (positive) limit, and lower negative) limit, respectively, of the depth of focus. For simplicity, these variables are expressed as integer multiples of a fixed stepsize Δz. These focal variables are ordinarily given fixed values in eq. 1, in order that the equation can be solved as a pure LP using standard methods. This LP can then be embedded in an outer search loop that finds optimal values for the focal variables. An alternative approach may optionally be used in the present invention, as will be discussed below.
The optimal intensity that should be given to each source pixel in the collection of possible choices is represented as a list of unknowns. Following standard practices, a vector notation can be used for this list in which the optimal intensities of the different pixels are tabulated like the components of a vector. In Eq. 1, units for these source intensities are selected such that the maximum possible value that the illuminator can provide in any pixel is normalized to 1. The list of unknown pixel intensities is written as {right arrow over (s)}, but eq. 1 uses a re-scaled list  that is related to the desired solutions {right arrow over (s)} according to the following formula:
                              s          j                =                            ⁢                                                    Min                k                            ⁡                              [                                                      S                                          MAX                      ,                      k                                                                                        ]                                      .                                              (                  eq          .                                          ⁢          2                )            
In lithographic applications, it is usually preferable to define process window in terms of percentage or fractional variations, rather than absolute variations. The integrated fractional exposure latitude can be optimized by using the scaled {right arrow over (s)} variables rather than the integrated absolute dose or intensity latitudes.
It is convenient to use units for the sj such that the physical limits on their attainable values are scaled to a 0-to-1 range. However in particular problems the source intensities can have other more specialized limits imposed upon them. Towards that end, the parameters SMax,j in eqs. 1 and 2 represent problem-specific limits that can be imposed on the intensity of each j-th source pixel. Each SMax,j should always be 1 or less. Each SMax,j is 1 if the full range of deliverable pixel intensity is to be considered. However, the SMax,j parameters allow stronger restrictions to optionally be placed on some or all of the pixels. For example, a resemblance of the solution to some reference source of interest can be enforced by limiting the intensities of those solution pixels that are switched-off in the reference source.
In addition, it is often inconvenient to maintain the 0-to-1 pixel intensity scale when the solution is ported to other modules for subsequent lithographic analysis and optimization. For example, the source pixels will typically have different areas, and larger pixels will usually be able to deliver proportionately more light. Eq. 1 uses the symbol {right arrow over (p)} to denote the list of maximum possible intensities in un-normalized form that the various pixels can deliver. Often pj would simply be the area of the j-th pixel, e.g., in units where the full pupil area is 1. Each source pixel is typically given the shape of a symmetric quadrupole in order to avoid skew through focus. By symmetry the shape of the pixel-pole in one quadrant of the illumination pupil suffices to determine the pixel shape in the other three quadrants. By convention, pixel properties are often described in terms of single-quadrant parameters only.
The desired properties of the image intensity distribution are specified using discrete sample points, i.e., evaluation points or positions in the image where the metric to be optimized is measured or evaluated in the simulated image. The density of the simulated image sufficiently exceeds the lens resolution that the sampled intensity adequately represents the character of the image. For example, certain classes of sample points can be used to map out the nominally bright and dark regions of the image. Other classes define the boundaries of the acceptable band of positions in which each feature edge can be allowed to print (tolerance bands). During source optimization, the mask or masks that form the image are usually fixed. In the image plane, the intensity at a given point can be written as a sum of contributions from each unknown source element, i.e., as {right arrow over (s)}·{right arrow over (I)}, where {right arrow over (I)} is a list of the intensities that each pixel would provide at the given image point if fully switched on. (The prime mark that appears on {right arrow over (I)}′ in eq. 1 is issued below.) The intensity contributions from different elements add incoherently. For a given mask, each element of {right arrow over (I)}′ thus expresses as a numerical value the imaging relationship that exists between the intensity of the corresponding source pixel and the intensity at a particular point in the image. These numerical values can be calculated using well known image simulation methods.
Different target shapes might be desired for the printed feature in different resist regions D. For example, in advanced damascene technologies, the 3D target shape might have different 2D cross-sections that reflect the desire to print different 2D target shapes at different depths in the resist. Since the image shape is defined by sample points, such 3D shape requirements entail the use of separate categories of sample points that are specific to different regions D. To distinguish bright and dark sample points in different regions, a notational convention can be used where the generic subscript index i is written as i(u, D) (for bright) or i(v, D) (dark), in a given region D, in which u and v are therefore indices which represent different bright and dark sample points. In a given region D, these indices suffice to identify an overall sample point index i. Use of distinct letters u and v is adopted to clearly distinguish the bright or dark status of sample points in the interior of features. The number of sample points in different D regions may vary, leading to changing integer limits on u and v.
Though this option is mentioned here, explicit identification of depth regions are avoided for simplicity in the discussion that follows. It will be clear to those skilled in the art how they may be included.
Fractional (percentage) objectives, like integrated fractional exposure latitude, typically require the specification of a particular “anchor feature” to serve as the normalizing reference. Generally, the printed edge of some critical dimension (“CD”) is chosen as the anchor feature in the prior art problem. By convention, a D=1 region (e.g. depth position 1) is designated as the region which contains the anchor feature. For brevity, eq. 1 then makes use of an “effective intensity” that takes into account the varying sensitivity of the resist according to the formula:{right arrow over (I)}′≡t(1){right arrow over (I)}/t(n)  (Eq. 3)
By using this method, the intensity can be generalized to incorporate a photoresist mean time for modualtion transfer function (“MTF”) that accounts for effects like resist diffusion. This generalized or effective intensity will be referred to simply as the intensity for brevity. When applied in the image plane, the term “intensity” can be assumed to refer to the generalized or effective intensity unless otherwise stated.